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Let (x_1,x_2) and (y_1,y_2,y_3) be two sets of complex numbers linearly independent over the rationals. Then at least one of ...

A Lie algebra g is solvable when its Lie algebra commutator series, or derived series, g^k vanishes for some k. Any nilpotent Lie algebra is solvable. The basic example is ...

The spherical distance between two points P and Q on a sphere is the distance of the shortest path along the surface of the sphere (paths that cut through the interior of the ...

A sliver of the surface of a sphere of radius r cut out by two planes through the azimuthal axis with dihedral angle theta. The surface area of the lune is S=2r^2theta, which ...

The combination of a central dilation and a rotation about the same center. However, the combination of a central dilation and a rotation whose centers are distinct is also a ...

Stanley and Wilf conjectured (Bona 1997, Arratia 1999), that for every permutation pattern sigma, there is a constant c(sigma)<infty such that for all n, ...

Let V be a real vector space (e.g., the real continuous functions C(I) on a closed interval I, two-dimensional Euclidean space R^2, the twice differentiable real functions ...

In the process of attaching a k-handle to a manifold M, the boundary of M is modified by a process called (k-1)-surgery. Surgery consists of the removal of a tubular ...

The set of elements belonging to one but not both of two given sets. It is therefore the union of the complement of A with respect to B and B with respect to A, and ...

Let A denote an R-algebra, so that A is a vector space over R and A×A->A (1) (x,y)|->x·y, (2) where x·y is vector multiplication which is assumed to be bilinear. Now define ...